基于Kriging代理模型的分数阶粘弹性反问题的数值求解
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摘要
分数阶导数粘弹性本构参数的识别具有重要的工程应用背景与理论探讨价值。在分数阶粘弹性反问题的求解中,由于敏度分析较为困难,一般需利用蚁群算法等智能类优化算法进行计算,而这类算法需多次求解正问题,计算成本颇大。本文将Kriging代理模型技术应用于分数阶粘弹性本构参数的识别,论文的主要学术贡献是：
1.提出了一种基于Kriging代理模型的求解分数阶粘弹性正问题的数值求解方法。利用拉丁超立方采样技术选取样本点,借助有限元和有限差分技术计算样本点的位移响应,从而建立Kriging代理模型。并针对问题的时间相关性,提出了构建Kriging代理模型时域分段建模的策略。通过均质和区域非均质两个分数阶粘弹性算例,对所提方法和策略的计算精度和计算效率进行了测试。测试分析表明,代理模型可提供合用的计算精度,当问题规模较大且正问题需要多次求解时,有望显著降低计算时间。
2.提出了一种基于Kriging代理模型的求解分数阶粘弹性反问题的数值求解方法。以已构建的Kriging代理模型作为正问题求解模型,以一种改进的连续域蚁群算法作为反问题求解模型。通过均质和区域非均质两个分数阶粘弹性参数组合反演的算例,对所提方法的反演计算精度和计算效率进行了测试,并探讨了数据噪音对反演结果的影响。计算结果表明,所提方法在保证了一定的反演计算精度的同时,能够显著地缩短反演计算时间。
本文的研究工作有望为提高分数阶粘弹性反问题与优化问题的计算效率提供有价值的参考和依据。
It has important application background in engineering and research value in theory to identify constitutive parameters of fractional viscoelastic problems. The Intelligent Optimization Algorithms, such as Ant colony Algorithm, are employed to solve the inverse fractional viscoelastic problems due to the difficulty of sensitivity analysis, but they are required to solve direct problem repeatedly which lead to a considerable amount of computing cost. In this paper, the Kriging surrogate model is applied to identify constitutive parameters of fractional viscoelastic problems, and the main academic contributions of this paper are:
1. Developing a Kriging surrogate model based numerical method to solve direct fractional viscoelastic problems. The Kriging surrogate model is set up by making use of the displacement solution given by FEM and FDM at sample points, and sample points are selected via a Latin Hypercube sampling technique. The paper proposes a strategy of piecewise modeling in time domain for constructing Kriging surrogate model due to the temporal correlations. The computing accuracy and efficiency of the proposed method and strategy are tested via homogeneous/regionally inhomogeneous fractional viscoelastic numerical examples. The test indicates that the surrogate model is able to provide a reasonable computing accuracy, and is hopefully to save considerable amount of computing expense for large scale direct fractional viscoelastic problem which are required to solve repeatedly.
2. Developing a Kriging surrogate model based numerical method to solve the inverse fractional viscoelastic problems. The established Kriging surrogate model is used as the solution model of direct problem and an improved Ant colony algorithm is employed as the solution model of inverse problem. The computing accuracy and efficiency of the proposed method are tested via numerical examples of combined identification for fractional viscoelastic parameters in cases of homogeneity and regionally inhomogeneity, and the effects of date noise are considered. The computing results demonstrate that the amount of computing expense is dramatically reduced as reasonable computing accuracy of inversion is maintained.
This paper is hopefully to provide a valuable reference for improving computing efficiency of inverse/optimization fractional viscoelastic problems.
1.提出了一种基于Kriging代理模型的求解分数阶粘弹性正问题的数值求解方法。利用拉丁超立方采样技术选取样本点,借助有限元和有限差分技术计算样本点的位移响应,从而建立Kriging代理模型。并针对问题的时间相关性,提出了构建Kriging代理模型时域分段建模的策略。通过均质和区域非均质两个分数阶粘弹性算例,对所提方法和策略的计算精度和计算效率进行了测试。测试分析表明,代理模型可提供合用的计算精度,当问题规模较大且正问题需要多次求解时,有望显著降低计算时间。
2.提出了一种基于Kriging代理模型的求解分数阶粘弹性反问题的数值求解方法。以已构建的Kriging代理模型作为正问题求解模型,以一种改进的连续域蚁群算法作为反问题求解模型。通过均质和区域非均质两个分数阶粘弹性参数组合反演的算例,对所提方法的反演计算精度和计算效率进行了测试,并探讨了数据噪音对反演结果的影响。计算结果表明,所提方法在保证了一定的反演计算精度的同时,能够显著地缩短反演计算时间。
本文的研究工作有望为提高分数阶粘弹性反问题与优化问题的计算效率提供有价值的参考和依据。
It has important application background in engineering and research value in theory to identify constitutive parameters of fractional viscoelastic problems. The Intelligent Optimization Algorithms, such as Ant colony Algorithm, are employed to solve the inverse fractional viscoelastic problems due to the difficulty of sensitivity analysis, but they are required to solve direct problem repeatedly which lead to a considerable amount of computing cost. In this paper, the Kriging surrogate model is applied to identify constitutive parameters of fractional viscoelastic problems, and the main academic contributions of this paper are:
1. Developing a Kriging surrogate model based numerical method to solve direct fractional viscoelastic problems. The Kriging surrogate model is set up by making use of the displacement solution given by FEM and FDM at sample points, and sample points are selected via a Latin Hypercube sampling technique. The paper proposes a strategy of piecewise modeling in time domain for constructing Kriging surrogate model due to the temporal correlations. The computing accuracy and efficiency of the proposed method and strategy are tested via homogeneous/regionally inhomogeneous fractional viscoelastic numerical examples. The test indicates that the surrogate model is able to provide a reasonable computing accuracy, and is hopefully to save considerable amount of computing expense for large scale direct fractional viscoelastic problem which are required to solve repeatedly.
2. Developing a Kriging surrogate model based numerical method to solve the inverse fractional viscoelastic problems. The established Kriging surrogate model is used as the solution model of direct problem and an improved Ant colony algorithm is employed as the solution model of inverse problem. The computing accuracy and efficiency of the proposed method are tested via numerical examples of combined identification for fractional viscoelastic parameters in cases of homogeneity and regionally inhomogeneity, and the effects of date noise are considered. The computing results demonstrate that the amount of computing expense is dramatically reduced as reasonable computing accuracy of inversion is maintained.
This paper is hopefully to provide a valuable reference for improving computing efficiency of inverse/optimization fractional viscoelastic problems.
引文
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[3]杨海天,赵潇.蚁群算法求解二维分数阶粘弹性参数反问题[C].郑州：中国力学学会,2009.
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[7]段海滨.蚁群算法原理及其应用[M].北京：科学出版社,2005.
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[10]Podlubny I. Fractional Differential Equations[M]. New York:Academic Press,1999.
[11]Rossikhin Y A, Shitikova M V. Analysis of damped vibrations of linear viscoelastic plates with danmping modeled with fractional derivatives [J]. Signal Processing, 2006,86(10):2703-2711.
[12]Cortes F, Elejabarrieta M J. Finite element formulations for transient dynamic analysis in structural systems with viscoelastic treatments containing fractional derivative models [J]. International Journal for Numerical Methods in Engineering, 2007,69(10):2173-2195.
[13]陈立群,程昌钧.分数导数型本构关系描述粘弹性梁的振动分析[J].力学季刊,2001,22(4)：512-516.
[14]张为民,张淳源,张平.考虑老化的混凝土粘弹性分数导数模型[J].应用力学学报,2004,21(1)：1-5.
[15]Galue L, Kalla S L, Al-Saqabi B N. Fractional extensions of the temperature field problems in oil strata [J]. Applied Mathematics and Computation,2007,186(1): 35-44.
[16]同登科,石丽娜.多孔介质中粘弹性液体广义流动分析[J].水动力学研究与进展A,2004,19(6)：695-701.
[17]Li C, Chen G. Chaos in the fractional order Chen system and its control [J]. Chaos, Solitons and Fractals,2004,22(3):549-554.
[18]Yin Y B, Zhu K Q. Oscillating flow of a viscoelastic fluid in a pipe with the fractional Maxwell model [J]. Applied Mathematics and Computation,2006,173(1): 231-242.
[19]El-Misiery A E M, Ahmed E. On a fractional model for earthquakes [J]. Applied Mathematics and Computation,2006,178(2):207-211.
[20]Carcione J M, Cavallini F, Mainardi F, Hanyga A,张志中,安镇文,丁志峰.应用分数阶微分对地震波稳恒Q值的时间域模拟[J].世界地震译丛,2005,(1)：51-58.
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